I'm looking for more information on the matrix that the Wronskian acts on to provide information about a set of solutions to an ODE.
So far I understand that:
- The Wronskian comes from Cramer's rule (at least that is the explanation I see most of the time).
- If the Wronskian is not equal to zero, then the set of solutions is linearly independent.
- If the Wronskian is equal to zero, that the set of solutions could still be linearly independent, but likely isn't.
- If the determinant of a matrix is equal to nonzero, then the matrix is linearly independent, it can be inverted and all that jazz.
What I'm confused on:
- Why did we choose the derivatives of the function in the matrix inside of the Wronskian?
- I've seen some... lame proofs that the derivatives of differentiable functions should be "linearly independent" to each other. I can easily think of functions that are differentiable that don't follow that rule (like $e^x$).
- (This might be a stupid question): If the columns of the matrix are not linearly independent then isn't the entire matrix linearly dependent? Is this where our problems of the Wronskian doesn't guarantee linear dependence arise from?